A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Showing that a Markov jump process is a Feller-Dynkin process

Let $E$ be a countable state space with $\sigma$-algebra $2^E$ and $X_t$ a Markov jump process with transition function $$P_t(x,y) = \sum_{n=0}^\infty e^{-\lambda t}\frac{(\lambda ...
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1answer
23 views

Poisson Process. Expected time of three fishermen catching at least three fish.

Three fishermen are fishing, we model the fishing as a Poisson Process of rate $2.5$ fish/hour. The fishermen leave only when all of them have caught at least 3 fish, we call this leaving time $T$. ...
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15 views

A Quick Question on Filtrations and Stopping Times

Let $(\Omega,\mathcal{F},P)$ be a probability space with filtration $\{\mathcal{F}_t:t\geq0\}$. Define $\mathcal{F}_{t+}=\bigcap\limits_{s>t}\mathcal{F}_s$. If $\tau$ is a random time and $\forall ...
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23 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
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29 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even tough its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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1answer
22 views

Does the random variable $f(\tau)M_\tau$, where $M$ is a martingale and $\tau$ is a stopping time, have zero expectation?

Suppose that $M:=\{M_t\}_{t\geq0}$ is a martingale adapted to some filtration $\mathcal{F}:=\{\mathcal{F}_t\}_{t\geq0}$ with $M_0\equiv0$ and that $\tau$ is an $\mathcal{F}_t$-stopping time. Suppose ...
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1answer
23 views

Existence of steady state distribution for finite state Markov chains

Let's assume a Markov chain has 2 recurrent classes and a transient state from which we can go to either of the recurrent classes. If one of those recurrent classes is periodic, would it effect the ...
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1answer
32 views

What is the distribution of a stochastic process?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be a random variable on ...
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11 views

Question about Weiner process [on hold]

Suppose $X(t)$ is a Weiner process. How would I be able to obtain the distribution of $\sup_t (X(t)-2t)$?
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19 views

Scaling Gaussian process

If $X\sim N(0,I)$, then $VX \sim N(0,VV^\top)$. Consider the Gaussian process $Y\sim \mathcal{G}(0,k(t,t')),t,t'\in[0,1]$, that is any finite-dimensional distribution of $Y$ is a multivariate normal. ...
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20 views

Why is the expectation of essential supremum equal the supremum of expectations

Let $\{X_i\}$ be a sequence sequence of nonnegative r.v. which has the lattice property. This implies that there is a sequence of indices $\{i_n\}$ so that $\{X_{i_n}\}$ is nondecreasing and ...
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1answer
18 views

Can an indicator function be a valid Radon Nikodym derivative?

Take a process $X_t$ defined on a canonical space with probability $\mathbb{P}$. Can the indicator function $\mathbb{1}_{X_t< U}$ be a Radon Nikodym derivative? That is can we have a measure ...
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1answer
40 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
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27 views

how to prove this function is a probability measure in $U_B$

Let $(\Omega, U, P)$ be a probability space. and $B\in U$, $P(B)\gt 0$ $U_B =\{A: A=B\cap C, C\in U\}$ its class in $\Omega$ is a $\sigma$-algebra and $P_B : U_B \to \Bbb R$ $A \to ...
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2answers
19 views

Transition probability matrix

In the article here it had this question. A walker moves on two positions a and b. She begins at a at time 0, and is at a next time as well. Subsequently, if she is at the same position for two ...
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1answer
44 views

Long term probability in Markov Chains

I was practicing some questions on transition probability matrices and I came up with this question. You have 3 coins: A (Heads probability 0.2),B (Heads probability 0.4), C (Heads probability ...
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16 views

Preparation for research in statistical inference for stochastic processes [on hold]

I am interested in building capacity to do research (and ultimately building a career) in statistical inference for continuous time stochastic processes. I am seeking advice on how best to go about ...
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12 views

Ross Intro to Probability Models--Example 4.4

Can someone explain to me please how we derived the Transition Matrix? Why we decided to put $P_{00} =0.7$ and $1 - P_{00} = P_{02}$. I just don't see it the way Ross defined the different states. ...
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1answer
41 views

Separable generating function of a pair of dependent discrete random variables

Independence is sufficient but not necessary for the generating function of the sum of two random variables to be the product of their individual generating functions. I am trying to come up with an ...
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1answer
23 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
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1answer
36 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
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1answer
24 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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10 views

Cholesky decomposition of positive semi-definite covariance matrix

I am trying to simulate a column vector of random complex variables, $\boldsymbol{x}$, which has a has a given covariance matrix: $$ \boldsymbol{C}=E\left[\boldsymbol{x}\boldsymbol{x}^{*}\right] $$ ...
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27 views

what is determinantal process?

Would anyone please explain what does this mean? A random point process $P$ on a discrete base set $Y = \{1,\ldots,N\}$ is a probability measure on the set $2^Y$ of all subsets of $Y$. Let $K$ ...
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1answer
34 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
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1answer
74 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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1answer
32 views

Strong Markov Property and Product of Expectations

Let $(B_{t})_{t\geq0}$ be a Brownian motion and let $\tau=\inf\left\{ t\geq0:B_{t}\leq-4\right\} $ be a stopping time. Then the strong Markov property ensures that e.g. $A:=\left\{ ...
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1answer
51 views

What is the integral of a family of diffusion processes? [on hold]

Let $S$ be an infinite subset of $[0,1]$. For all $s \in S$, let W_s(t) be a standard Wiener process. Definite P(s)_t = \mu(P,s,t) dt + \sigma(P,s,t) dW^s_t Can we characterize? $$F_t= \int_S P(s)_t ...
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Informal Introduction to Cox Processes

Can anyone recommend a good introduction (preferably not terribly formal) to Cox processes? (Inhomogenous Poisson processes with a parameter that is also stochastic.) This introduction to spatial ...
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1answer
26 views

Show a random walk is transient

I was going through some problems related to Markov chains and I got stuck on this bit: We are given a random walk on $Z$, defined by the transition matrix $p_{i,i+1}=p$ and $p_{i,i-1}=1-p$. How to ...
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1answer
18 views

Limiting distribution of a Markov chain?

I have the problem below. There are n identical machines. They are all operational at time 0. The lifetime of each one is an exponential random variable with rate L. There are r repairmen (1 ≤ r ≤ ...
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21 views

$2$-dimensional density of Brownian bridge?

I know that a $1$-dimensional Brownian bridge $B(t)$ just follows a normal distribution with mean $0$ and variance $t(1-t)$. But how do I compute the 2-dimensional density? I mean, $\{B(s), B(t)\}$ ...
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25 views

Find $dE(x)$ where $dx$ is defined [on hold]

I'm confused with this example? Could someone explain this? $dx$ is defined as following: $dx = -kxdt + σdw$ why $dE(x) = -kE(x)dt + σ E(dw)$ ?
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2answers
42 views

Ito Differential Equation example [on hold]

Could someone explain Ito through an example as following? How to use Ito differential equation to find $dy$ , where $y = e^{w(t)}$
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0answers
18 views

calculate the mean and covariance functions of the stochastic process $X$ [closed]

Let $X=\left(X_{t}\right)_{t\geq 0}$ a stochastic process given by: $X_{t}:=A\cos\left(\varphi+\lambda t \right)$ where $\lambda>0$ is a constant, $A$ and $\varphi$ are independent random ...
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If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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1answer
22 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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2answers
27 views

Insurance claims Poisson problem derving expected value and variance

If I have that claims arrive at an insurance company according to a Poisson process $\{N(t) : t \ge 0\}$ at a rate $\lambda > 0$ and $X_i$ denotes the claim size of the $ith$ claim. I assume that ...
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19 views

Examples of Non-Markov process with continuous time and finite set of states.

What is the best real world examples of non-Markov process with continuous time, but with finite set of states?
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33 views

Expectation of squared Ito integral

Let $\omega$ be a standard Brownian motion. How do you compute the expectation involving the square of an Ito integral: $ ...
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0answers
14 views

Prove that an operator from $L^2(\Omega;C(s,T;\mathbb R^n ))$ into itself is well defined

I need an help proving the following estimate. First, we fix the notation. Let $L^2(\Omega;C(s,T;\mathbb R^n ))$ be the set of continuous and adapted processes $\{X_t:t\in [s,T]\}$ (valued in ...
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Definition of rough path

There are many books and notes on the rough path theory. A rough path is defined as an ordered pair $(X, \mathbb X)$, where $X$ is a path mapping from $[0, T]$ to some Banach space $V$ and $\mathbb X: ...
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Proving that a process is class DL

Let $(X_{t})$ be a stochastic process with $X_{t}\sim\mathcal{N}(\xi_{t},\sigma_{t}^{2})$ where $\xi_{t}\downarrow0$ and $\sigma_{t}^{2}\uparrow1/2$. What would be the most straightforward way ...
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22 views

Is a linear random walk with jump recurrent?

Let $\lambda_0=10^5$ or any other large integer. Define the recursive "process": $\lambda_t=\text{sample from a Poisson distribution with mean }\lambda_{t-1}$. Is this process recurrent? I mean, after ...
2
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1answer
25 views

Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
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1answer
24 views

Poisson process Probabilities

If I assume that $\{N(t)=: t \ge 0\}$ is a Poisson process with intensity $\lambda$. For $0<s<t$, how would I find the $\Pr\{N(t)>N(s)\}$?
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31 views

Markov Chain States

Given a MC with five states $\{1,2,3,4,5\}$ and transition matrix \begin{bmatrix} 0.5& 0.5 & 0 & 0 & 0 \\ 0.75 & 0.25 & 0 & 0 & 0 \\ 0 & 0.25 & ...
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1answer
24 views

dependent “time change” of a.s. convergent random variables

Let $(X_n)$ be a sequence of random variables, s.t. $\frac{X_n}{n^p}\to X$ a.s. for some $p>0$. Now let $(Y_t)$ be a discrete stochastic process, s.t. $\frac{(Y_t)^p}{t}\to Y>0$ a.s. We only ...
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1answer
27 views

Show that $((N_t-t)^2-t)_{t \geq 0}$ is a martingale for a Poisson process $(N_t)_{t \geq 0}$

I am asked to show that if $N$ is a poisson process of intensity $1$, then: $X_t=N_t-t$ is a martingale. $X_t^2-t$ is a martingale. I have done the first part easily, using independence of ...
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1answer
36 views

Geometric Brownian Motion [closed]

I am new there. How can I calculate following expected value: $$E[X(s)\times X(t)]$$ where $X$ is Geometric Brownian Motion, i.e. $X(t) = exp[(\mu - 0.5\cdot \sigma^2)t + \sigma\cdot W(t)]$ ...